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Theorem psubclsubN 38406
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclsub.s 𝑆 = (PSubSpβ€˜πΎ)
psubclsub.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubclsubN ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 ∈ 𝑆)

Proof of Theorem psubclsubN
StepHypRef Expression
1 eqid 2737 . . 3 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
2 psubclsub.c . . 3 𝐢 = (PSubClβ€˜πΎ)
31, 2psubcli2N 38405 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋)
4 eqid 2737 . . . . . . 7 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
54, 1, 2psubcliN 38404 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋))
65simpld 496 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
7 psubclsub.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
84, 7, 1polsubN 38373 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) ∈ 𝑆)
96, 8syldan 592 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) ∈ 𝑆)
104, 7psubssat 38220 . . . 4 ((𝐾 ∈ HL ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) ∈ 𝑆) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
119, 10syldan 592 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
124, 7, 1polsubN 38373 . . 3 ((𝐾 ∈ HL ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝑆)
1311, 12syldan 592 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝑆)
143, 13eqeltrrd 2839 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  β€˜cfv 6497  Atomscatm 37728  HLchlt 37815  PSubSpcpsubsp 37962  βŠ₯𝑃cpolN 38368  PSubClcpscN 38400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18185  df-poset 18203  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-psubsp 37969  df-pmap 37970  df-polarityN 38369  df-psubclN 38401
This theorem is referenced by:  pclfinclN  38416
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